Completing Wheeler Automata

We consider the problem of embedding a Wheeler Deterministic Finite Automaton (WDFA, in short) into an equivalent complete WDFA, preserving the order of states and the accepted language. In some cases, such a complete WDFA does not exist. We say that a WDFA is Wheeler-complete (W-complete, in short) if it cannot be properly embedded into an equivalent WDFA. We give an algorithm that, given as input a WDFA A, returns the smallest W-complete DFA containing A: it is called the minimal W-completion of A. We derive some interesting applications of this algorithm concerning the construction of a WDFA for the union and a WDFA for the complement of Wheeler languages.